Let be a set of simple roots for g. Then the associated Cartan matrix is the matrix with entries < , / <, >. The entries of the Cartan matrix are 0, 1, -1 or 2. The Cartan matrix is independent of the choice of Cartan subalgebra h but is dependent upon the ordering of the simple roots in

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The Cartan matrix is the fundamental invariant for semi-simple Lie algebras over C -- two complex semi-simple Lie algebras are isomophic if and only if their Cartan matrices are the same, modulo a permutation of the vectors in the Cartan subalgebra. The command CartanMatrixToStandardForm will transform a given Cartan matrix to a standard form.

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The Cartan matrix encodes the re-construction of the root system of the Lie algebra from its simple roots. See PositiveRoots .

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The information contained in the Cartan matrix is also encoded in the Dynkin diagram of the Lie algebra.

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The first calling sequence calculates the Cartan matrix of a Lie algebra from a set of simple roots and a root space decomposition.

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The second calling sequence displays the standard form of the Cartan matrix for each possible root type of a simple Lie algebra.

Examples

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Example 1.

We use the command SimpleLieAlgebraData to obtain the Lie algebra data for the Lie algebra . This is the 15-dimensional Lie algebra of trace-free, skew-Hermitian matrices

We suppress the output of this command which is a lengthy list of structure equations.

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Initialize this Lie algebra -- the basis elements are given the default labels